Finite Group Databasebeta

← #89608⋊C3 #13051739

Subgroups — #89608⋊C3 #13051739

Order 1344.

248 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1131⋊C3C211yesmaximal, characteristic, complemented
2C2×C8×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3C4×C4⋊AGL(1,𝔽7)C211yesmaximal, characteristic
4ΓL(1,𝔽8)×C4⋊C8C211yesmaximal, characteristic, complemented
5(C2×AGL(1,𝔽7))⋊C8C211yesmaximal, characteristic, complemented
6(C2×AGL(1,𝔽7))⋊C8C211yesmaximal, characteristic, complemented
7#6377⋊C3C211yesmaximal, characteristic, complemented
8#89608C311yesmaximal, characteristic, complemented, Hall
9C2×C4⋊AGL(1,𝔽7)C411yescharacteristic
10C2×C4×AGL(1,𝔽7)C2211yescharacteristic
11C2×C4×AGL(1,𝔽7)C2211yescharacteristic
12C2×C8×ΓL(1,𝔽8)C2211yescharacteristic
13C2×C8×ΓL(1,𝔽8)C2211yescharacteristic
14C42×ΓL(1,𝔽8)C2211yescharacteristic
15(C2×AGL(1,𝔽7))⋊C4C411yescharacteristic
16(C2×AGL(1,𝔽7))⋊C4C411yescharacteristic
17C4⋊(ΓL(1,𝔽8)⋊C4)C411yescharacteristic
18C2×ΓL(1,𝔽8)⋊C8C2211yescharacteristic
19C2×ΓL(1,𝔽8)⋊C8C2211yescharacteristic
20#1131⋊C3—22no—
21C8×AGL(1,𝔽7)—22no—
22C2×#1131C2×C311yescharacteristic, complemented
23C2×C8×D7C2×C311yescharacteristic, complemented
24C4×D28C2×C311yescharacteristic
25C7×C4⋊C8C2×C311yescharacteristic, complemented
26(C2×D7)⋊C8C2×C311yescharacteristic, complemented
27(C2×D7)⋊C8C2×C311yescharacteristic, complemented
28#6377C2×C311yescharacteristic, complemented
29C3×#435—71nomaximal, Hall
30C22×AGL(1,𝔽7)C2×C411yescharacteristic
31C22×AGL(1,𝔽7)C2×C411yescharacteristic
32C2×C4×ΓL(1,𝔽8)C2×C411yescharacteristic
33C2×C4×ΓL(1,𝔽8)C2×C411yescharacteristic
34C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
35C2×ΓL(1,𝔽8)⋊C4C2×C411yescharacteristic
36C2×ΓL(1,𝔽8)⋊C4C2×C411yescharacteristic
37C4⋊AGL(1,𝔽7)—22no—
38C4×AGL(1,𝔽7)—21no—
39C4×AGL(1,𝔽7)—21no—
40C4×AGL(1,𝔽7)—21no—
41C4×AGL(1,𝔽7)—21no—
42C8×ΓL(1,𝔽8)—21no—
43C8×ΓL(1,𝔽8)—21no—
44ΓL(1,𝔽8)⋊C8—21no—
45ΓL(1,𝔽8)⋊C8D412yescomplemented
46C2×D28C3×C411yescharacteristic
47C2×C4×D7C22×C311yescharacteristic
48C2×C4×D7C22×C311yescharacteristic
49C2×C7×C8C22×C311yescharacteristic
50C2×C7×C8C22×C311yescharacteristic
51C42×C7C22×C311yescharacteristic
52(C2×D7)⋊C4C3×C411yescharacteristic
53(C2×D7)⋊C4C3×C411yescharacteristic
54C4⋊Dic7C3×C411yescharacteristic
55C2×C7⋊C8C22×C311yescharacteristic
56C2×C7⋊C8C22×C311yescharacteristic
57#1131—22no—
58C8×D7—22no—
59C2×C3×M16—71no—
60C22×C3×C8—71no—
61C3×C4×D4—71no—
62C3×C4⋊C8—71no—
63C3×C22⋊C8—71no—
64C3×C22⋊C8—71no—
65C3×#97—71no—
66C22×ΓL(1,𝔽8)C22×C411yescharacteristic
67C2×AGL(1,𝔽7)—21no—
68C2×AGL(1,𝔽7)—21no—
69C2×AGL(1,𝔽7)—21no—
70C2×AGL(1,𝔽7)—21no—
71C2×AGL(1,𝔽7)—21no—
72C2×AGL(1,𝔽7)M1612yescomplemented
73C4×ΓL(1,𝔽8)—21no—
74C4×ΓL(1,𝔽8)—21no—
75C4×ΓL(1,𝔽8)C2×D411yescharacteristic
76C4×ΓL(1,𝔽8)#4111yescharacteristic
77ΓL(1,𝔽8)⋊C4—21no—
78ΓL(1,𝔽8)⋊C4—21no—
79#435—71noSylow
80C22×D7C2×C3×C411yescharacteristic
81C22×D7C2×C3×C411yescharacteristic
82C2×C4×C7C2×C3×C411yescharacteristic
83C2×C4×C7C2×C3×C411yescharacteristic
84C2×C4×C7C23×C311yescharacteristic
85C2×Dic7C2×C3×C411yescharacteristic
86C2×Dic7C2×C3×C411yescharacteristic
87D28—22no—
88C4×D7—21no—
89C4×D7—21no—
90C4×D7—21no—
91C4×D7—21no—
92C7×C8—21no—
93C7×C8—21no—
94C7⋊C8—21no—
95C7⋊C8C3×D412yescomplemented
96C2×C3×D4—71no—
97C22×C3×C4—71no—
98C22×C3×C4—71no—
99C3×M16—142no—
100C2×C3×C8—71no—
101C2×C3×C8—71no—
102C2×C3×C8—71no—
103C2×C3×C8—71no—
104C2×C3×C8—142no—
105C3×C4⋊C4—71no—
106C3×C22⋊C4—71no—
107C3×C22⋊C4—71no—
108C3×C42—71no—
109C2×ΓL(1,𝔽8)C4×D411yescharacteristic
110C2×ΓL(1,𝔽8)C2×M1611yescharacteristic
111C2×ΓL(1,𝔽8)#13111yescharacteristic
112AGL(1,𝔽7)—22no—
113AGL(1,𝔽7)—41no—
114C2×M16—71no—
115C22×C8—71no—
116C4×D4—71no—
117C4⋊C8AGL(1,𝔽7)11yescharacteristic, complemented
118C22⋊C8—71no—
119C22⋊C8—71no—
120#97—71no—
121C22×C7C22×C3×C411yescharacteristic
122C2×D7—21no—
123C2×D7—21no—
124C2×D7—21no—
125C2×D7—21no—
126C2×D7—21no—
127C2×D7C3×M1612yescomplemented
128C4×C7—21no—
129C4×C7—21no—
130C4×C7C2×C3×D411yescharacteristic
131C4×C7C3×#4111yescharacteristic
132Dic7—21no—
133Dic7—21no—
134C23×C3—71no—
135C23×C3—71no—
136C3×D4—142no—
137C2×C3×C4—71no—
138C2×C3×C4—71no—
139C2×C3×C4—71no—
140C2×C3×C4—71no—
141C2×C3×C4—71no—
142C2×C3×C4—141no—
143C2×C3×C4—141no—
144C2×C3×C4—141no—
145C2×C3×C4—141no—
146C3×C8—72no—
147C3×C8—141no—
148C3×C8—141no—
149C3×C8—141no—
150ΓL(1,𝔽8)#43511yescharacteristic, complemented, Hall
151C2×D4—71no—
152C22×C4—71no—
153C22×C4—71no—
154M16—142no—
155C2×C8—71no—
156C2×C8—71no—
157C2×C8—142no—
158C2×C8C2×AGL(1,𝔽7)11yescharacteristic
159C2×C8C2×AGL(1,𝔽7)11yescharacteristic
160C4⋊C4—71no—
161C22⋊C4—71no—
162C22⋊C4—71no—
163C42C2×AGL(1,𝔽7)11yescharacteristic
164C2×C7C3×C4×D411yescharacteristic
165C2×C7C2×C3×M1611yescharacteristic
166C2×C7C3×#13111yescharacteristic
167D7—22no—
168D7—41no—
169C22×C3—72no—
170C22×C3—71no—
171C22×C3—141no—
172C22×C3—141no—
173C22×C3—141no—
174C22×C3—141no—
175C22×C3—141no—
176C3×C4—71no—
177C3×C4—71no—
178C3×C4—141no—
179C3×C4—141no—
180C3×C4—141no—
181C3×C4—141no—
182C23—71no—
183C23—71no—
184D4—142no—
185C2×C4—71no—
186C2×C4—71no—
187C2×C4—141no—
188C2×C4—141no—
189C2×C4—141no—
190C2×C4—141no—
191C2×C4C4×AGL(1,𝔽7)11yescharacteristic
192C2×C4C4×AGL(1,𝔽7)11yescharacteristic
193C2×C4C22×AGL(1,𝔽7)11yescharacteristic
194C8—21no—
195C8—21no—
196C8—72no—
197C8—141no—
198C7C3×#43511yescharacteristic, complemented, Sylow
199C2×C3—71no—
200C2×C3—71no—
201C2×C3—71no—
202C2×C3—142no—
203C2×C3—281no—
204C22—72no—
205C22—141no—
206C22—141no—
207C22—141no—
208C22—141no—
209C22—141no—
210C22C2×C4×AGL(1,𝔽7)11yescharacteristic
211C4—21no—
212C4—21no—
213C4—141no—
214C4—141no—
215C4D4×AGL(1,𝔽7)11yescharacteristic
216C4#6665911yescharacteristic
217C3—71noSylow
218C2—142no—
219C2—281no—
220C2#6456⋊C311yescharacteristic
221C2#6464⋊C311yescharacteristic
222C2M16×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×AGL(1,𝔽7)
commutator, socleC22×C7C22×C3×C4
FittingC7×C4⋊C8C2×C3
prime radicalC2×C4⋊AGL(1,𝔽7)C4
cosocleC2×C4×C7C23×C3
hypercenterC4⋊C8AGL(1,𝔽7)
hypocenterC7C3×#435
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#1131⋊C3C2semidirect
2. C2×C8×AGL(1,𝔽7)C2semidirect
4. ΓL(1,𝔽8)×C4⋊C8C2semidirect
5. (C2×AGL(1,𝔽7))⋊C8C2semidirect
6. (C2×AGL(1,𝔽7))⋊C8C2semidirect
7. #6377⋊C3C2semidirect
8. #89608C3semidirect
22. C2×#1131C2×C3semidirect
23. C2×C8×D7C2×C3semidirect
25. C7×C4⋊C8C2×C3semidirect
26. (C2×D7)⋊C8C2×C3semidirect
27. (C2×D7)⋊C8C2×C3semidirect
28. #6377C2×C3semidirect
45. ΓL(1,𝔽8)⋊C8D4semidirect
72. C2×AGL(1,𝔽7)M16semidirect
95. C7⋊C8C3×D4semidirect
117. C4⋊C8AGL(1,𝔽7)semidirect
127. C2×D7C3×M16semidirect
150. ΓL(1,𝔽8)#435semidirect
198. C7C3×#435semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#1131⋊C32
C2×C8×AGL(1,𝔽7)2
C4×C4⋊AGL(1,𝔽7)2
ΓL(1,𝔽8)×C4⋊C82
(C2×AGL(1,𝔽7))⋊C82
#6377⋊C32
#896083
C3×#4357